GATE 2022 CH – Question 40
Consider a bare long copper wire of 1 mm diameter. Its surface temperature is $T_s$ and the ambient temperature is $T_a$ ($T_s > T_a$). The wire is to be coated with a 2 mm thick insulation. The convective heat transfer coefficient is 20 W m$^{-2}$ K$^{-1}$. Assume that $T_s$ and $T_a$ remain unchanged. To reduce heat loss from the wire, the maximum allowed thermal conductivity of the insulating material, in W m$^{-1}$ K$^{-1}$, rounded off to two decimal places, is
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Correct answer: (D) 0.01
Explanation
The wire radius is $r_1 = 0.5$ mm and the outer radius of the insulation is $r_2 = 2.5$ mm. The resistance per unit length of the bare wire is $\frac{1}{2\pi r_1h}$. With the insulation it is $\frac{\ln(r_2/r_1)}{2\pi k} + \frac{1}{2\pi r_2h}$. The heat loss falls only if the second is larger: $\frac{\ln 5}{k} + \frac{1}{20 \times 0.0025} > \frac{1}{20 \times 0.0005}$, that is $\frac{1.609}{k} + 20 > 100$. So $k < \frac{1.609}{80} = 0.0201$... more precisely the limit is 0.0201, and the nearest option below it is 0.01.